Mathematical Analysis/Partial Differential Equations

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Mathematical Analysis/Partial Differential Equations

Lifting default for S ¹ -valued maps

Petru Mironescu

Université de Lyon, Université Lyon1, CNRS, UMR 5208, institut Camille-Jordan, bâtiment du Doyen Jean-Braconnier, 43, boulevard du 11 novembre 1918, 69200 Villeurbanne cedex, France

Received and accepted 30 July 2008 Available online 23 August 2008

Presented by Haïm Brezis

Abstract

Letϕ∈C^∞([0,1]^N,R). When 0< s <1,p1 and 1sp < N, theW^s,p-semi-norm|ϕ|W^s,p ofϕis not controlled by

|g|W^s,p, whereg:=e^ıϕ[J. Bourgain, H. Brezis, P. Mironescu, Lifting in Sobolev spaces, J. Anal. Math. 80 (2000) 37–86]. [This question is related to existence, forS¹-valued mapsg, of a liftingϕ as smooth as allowed byg.] In [J. Bourgain, H. Brezis, P. Mironescu, Lifting, degree, and distributional Jacobian revisited, Commun. Pure Appl. Math. 58 (2005) 529–551], the authors suggested that|g|W^s,pdoes control a weaker quantity, namely|ϕ|_W^s,p_+W^1,sp. Existence of such control is due to J. Bourgain and H. Brezis [J. Bourgain, H. Brezis, On the equation divY =f and application to control of phases, J. Amer. Math. Soc. 16 (2003) 393–426] when 1< p2,s=1/pand to H.-M. Nguyen [H.-M. Nguyen, Inequalities related to liftings and applications, C. R.

Acad. Sci. Paris, Ser. I 346 (17–18) (2008) 957–962] whenN=1,p >1 andsp1 or whenN2,p >1 andsp >1. In this Note, we establish existence of control for alls <1,p1 andN.To cite this article: P. Mironescu, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

©2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Résumé

Défaut de relèvement pour les applications à valeurs dans le cercle unité.Soitϕ∈C^∞([0,1]^N,R). Si 0< s <1,p1 et 1sp < N, alors la semi-norme|ϕ|W^s,p n’est pas contrôlée par|g|W^s,p, oùg:=e^ıϕ[J. Bourgain, H. Brezis, P. Mironescu, Lifting in Sobolev spaces, J. Anal. Math. 80 (2000) 37–86]. [Cette question est liée à l’existence, pour desgà valeurs dansS¹, de relèvementsϕaussi réguliers quegle permet.] Dans [J. Bourgain, H. Brezis, P. Mironescu, Lifting, degree, and distributional Jacobian revisited, Commun. Pure Appl. Math. 58 (2005) 529–551], il est conjecturé que|g|W^s,pcontrôle une quantité plus faible que|ϕ|W^s,p, plus spécifiquement|ϕ|_Ws,p+W^1,sp. L’existence d’un tel contrôle est due à J. Bourgain and H. Brezis [J. Bourgain, H. Brezis, On the equation divY=f and application to control of phases, J. Amer. Math. Soc. 16 (2003) 393–426] pour 1< p2 ets=1/pet à H.-M. Nguyen [H.-M. Nguyen, Inequalities related to liftings and applications, C. R. Acad. Sci. Paris, Ser. I 346 (17–18) (2008) 957–962] pourN=1,p >1 etsp1 ou pourN2,p >1 etsp >1. Dans cette Note, nous montrons l’existence d’un contrôle pour touts <1,p1 etN.Pour citer cet article : P. Mironescu, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

©2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

E-mail address:[email protected].

1631-073X/$ – see front matter ©2008 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.crma.2008.08.001

Version française abrégée

SoientC= [0,1]^N, 0< s <∞et 1p <∞.| |W^s,p désigne une semi-norme standard sur l’espace de Sobolev W^s,p(C); par exemple, pour 0< s <1 nous considérons la semi-norme de Gagliardo,

|u|W^s,p =

C²

|u(x)−u(y)|^p

|x−y|^N+sp dxdy 1/p

.

Si ϕ ∈C^∞(C,R)etg:=e^ıϕ, alors |∇ϕ| = |∇g|. En particulier,|g|_W^1,p = |ϕ|_W^1,p. Plus généralement, |g|W^s,p

contrôle|ϕ|W^s,p sis1, au sens où il existe une inégalité de la forme|ϕ|W^s,p F (|g|W^s,p)avecF croissante [3].

Ceci n’est plus forcément vrai si 0< s <1. Voici un exemple inspiré de [3] : siN 2 et si 0< s <1 etp sont tels que 1< sp < N, alors il existe une fonction ψ∈W^1,sp\W^s,p. Si on considère ϕ_ε:=ψ∗ρ_ε, avecρ noyau régularisant, alors|ϕ_ε|W^s,p → ∞(carψ /∈W^s,p), alors queg_ε:=e^ıϕε reste bornée dansW^1,sp∩L^∞(carψ∈W^1,sp) et donc dans W^s,p, grâce à l’inclusion de Gagliardo–NirenbergW^1,sp∩L^∞⊂W^s,p. Nonobstant la non inclusion W^1,1∩L^∞ ⊂W^s,psisp=1, on peut adapter cet exemple au cas oùsp=1 etN1. Plus généralement, sisp=1 ou 1< sp < N, alors on peut obtenir l’absence du contrôle à partir d’une fonction convenableψ∈W^s,p+W^1,sp.

Dans le cas particuliers=1/2 etp=2, J. Bourgain et H. Brezis [2] ont montré que le contre-exemple ci-dessus est essentiellement le seul. Leur résultat est que toute fonctionϕ se décompose commeϕ=ϕ₁+ϕ₂, où|ϕ₁|H^1/2 et

|ϕ₂|W^1,1sont contrôlées par|g|H^1/2. La preuve s’étend aux espacesW^1/p,pavec 1< p2 et donne la décomposition (1). Ce résultat a motivé le problème suivant [4,8] avec 0< s <1 et 1p <∞:

(D_s,p) Toutϕ∈C^∞(C,R)s’écritϕ=ϕ₁+ϕ₂, où|ϕ₁|W^s,pC|e^ıϕ|W^s,p etDϕ₂L^spC|e^ıϕ|^1/s_W^s,p.

Récemment, H.-M. Nguyen [10] a résolu ce problème lorsque p >1, s=1/petN =1 ; son argument s’applique aussi au casN2,p >1 etsp >1.

Le but de cette Note est d’annoncer le suivant :

Théorème 1.La décomposition(D_s,p)est valide pour tout0< s <1,p1etN.

Notons que la décomposition existe même sisp <1.

Idée de la preuve. On étendgà une applicationh:R^N→R², de sorte quehsoit Lipschitz, constante en dehors d’un compact,|h|3 et|h|W^s,p(R^N)C|g|W^s,p(C). On étend ensuitehàR^N×R^∗₊par la formulew(x, ε)=Π (h∗ρ_ε(x)).

Ici,Π∈C^∞(R²;R²)est telle queΠ (z)=z/|z|si|z|1/2, tandis queρ∈C₀^∞satisfaitρ0,

ρ=1, suppρ⊂ B(0,2)\B(0,1). Alors la conclusion du théorème est vérifiée parϕ_j données par

ϕ₁(x):= − ∞

0

w(x, ε)∧∂w

∂ε(x, ε)dε, ϕ₂:=ϕ−ϕ₁.

L’inégalité|ϕ₁|W^s,pC|h|W^s,p découle d’estimations standard¹pour les régulariséesh∗ρ_εd’une fonctionh∈W^s,p. Elle implique|ϕ₁|W^s,pC|g|W^s,p.

Pour estimerDϕ₂, le point de départ est l’identité²

Dϕ₂(x)= −2 ∞

0

∂

∂εw(x, ε)∧D_xw(x, ε)dε.

En adaptant une idée de [4], cette identité permet d’obtenir l’estimationDϕ₂L^spC|e^ıϕ|^1/s_Ws,p. 2

1 L’une des estimations utilisées dans la preuve est R^N_∞

0 ε^p−sp−1|D[h∗ρε](x)|^pdεdxC|h|^p_Ws,p, bien connue par les spécialistes [11], II.12.

2 Évidente, du moins formellement.

Le lecteur trouvera les preuves détaillées dans [9] ; entre autres, on y explique pourquoi notre décomposition est une sorte d’analogue continu de la décomposition trouvée dans [2].

1. Introduction

LetC= [0,1]^N, 0< s <∞and 1p <∞. We will denote by| |W^s,p a standard semi-norm on the Sobolev space W^s,p(C); e.g., when 0< s <1, we take the Gagliardo semi-norm

|u|W^s,p=

C²

|u(x)−u(y)|^p

|x−y|^N+sp dxdy 1/p

.

Letϕ∈C^∞(C,R)and setg=e^ıϕ. Since|∇ϕ| = |∇g|, we have|g|W^1,p= |ϕ|W^1,p. More generally,|g|W^s,pcontrols

|ϕ|W^s,p when s1, i.e., there is some nondecreasing F such that |ϕ|W^s,p F (|g|W^s,p)[3]. This need not hold when 0< s <1. Here is an example, essentially taken from [3]: letN2 and let 0< s <1, 1p <∞be such that 1< sp < N. By a Sobolev “nonembedding”, there is someψ∈W^1,sp\W^s,p. Letϕ_ε:=ψ∗ρ_ε and setg_ε= e^ıϕε, whereρ is a mollifier. Then|ϕ_ε|W^s,p → ∞(since ψ /∈W^s,p). On the other hand, g_ε:=e^ıϕε is bounded in W^1,sp∩L^∞ (sinceψ∈W^1,sp) and thus in W^s,p, by the Gagliardo–Nirenberg embeddingW^1,sp∩L^∞⊂W^s,p. Despite the nonembeddingW^1,1∩L^∞ ⊂W^s,pwhensp=1, one may easily adapt this example to the casesp=1 andN1. More generally, whensp=1 or 1< sp < N, one may prove lack of control with the help of an appropriate ψ∈W^s,p+W^1,sp.

In the special cases=1/2 andp=2, J. Bourgain and H. Brezis [2] proved that the above counter-example is essentially the only one. Their results asserts that each ϕ splits as ϕ=ϕ1+ϕ2, where |ϕ1|H^1/2 and|ϕ2|W^1,1 are controlled by|g|H^1/2. Their argument adapts steadily to the spacesW^1/p,pwhere 1< p2 and yields

If 1< p2, thenϕ=ϕ₁+ϕ₂

whereϕ_j∈C^∞(C), |ϕ₁|W^1/p,pC|g|W^1/p,p, |ϕ₂|W^1,1C|g|^p_W1/p,p. (1) This motivated the following open problem [4,8]:

(D_s,p) Eachϕ∈C^∞(C,R)splits asϕ=ϕ₁+ϕ₂, where|ϕ₁|W^s,p C|e^ıϕ|W^s,p andDϕ₂L^spC|e^ıϕ|^1/s_Ws,p. [Here, 0< s <1 and 1p <∞.] Very recently, H.-M. Nguyen [10] answered positively this problem whenp >1, s=1/pandN=1; his argument adapts to the case whereN2,p >1 andsp >1.

The main purpose of this Note is to announce the following:

Theorem 1.The decomposition(D_s,p)holds for each0< s <1,p1andN. Unlike the proofs in [2,10], our method applies to the casesp <1.

2. Heuristics of the proof of Theorem 1

In order to explain the main idea, we consider, for simplicity, maps defined onR^N which are constant at infin- ity (rather than maps defined on [0,1]^N). Assume first thatϕ has small amplitude oscillations, say|ϕ| 1. Then

|g−1| 1 and|ϕ|W^s,p ∼ |g|W^s,p. Thus, in this case, a convenient decomposition isϕ₁=ϕ andϕ₂=0. We next proceed as follows: we derive a formula forϕ. This formula givesϕ only whenϕ has small amplitude oscillations, but we may give it a meaning for eachϕ. In general (i.e., whenϕmay oscillate), we take this formula as the definition ofϕ1and simply letϕ2be the phase excess, i.e., we setϕ2=ϕ−ϕ1.

In order to obtain a tractable formula forϕ, we rely on the following remark: ifg=e^ıϕ, then there is no formula givingϕ in terms ofg, but there is one for Dϕ, since Dϕ=g∧Dg. We consider a smooth extension w:R^N× [0,+∞[ →S¹ofgsuch that

εlim→∞w(., ε)=const. (2)

A natural choice is w(x, ε)=Π (g∗ρ_ε(x)), whereΠ (z)=z/|z| andρ is a mollifier. This yields a smooth map wheneverg(and thusg∗ρ_ε) is close to 1. We may writew=e^ıψ, whereψ (x,0)=ϕ(x). Assuming that convergence in (2) is sufficiently fast, we than haveψ (x,∞)=Cand thus

ϕ(x)= −ψ (x, ε)|^ε_ε^=∞₌₀ +C=C− ∞

0

w(x, ε)∧∂w

∂ε(x, ε)dε. (3)

As explained in the next section, (3) gives the right definition ofϕ₁, provided we pick an appropriateρand we change slightly the definition ofΠ.

3. Sketch of the proof of Theorem 1

Let ϕ∈C^∞(C,R)and set g=e^ıϕ. We first extend gto a Lipschitz map h:R^N →R² such thath is constant outside[−1,2]^N,|h|3 and|h|W^s,p(R^N)C|g|W^s,p(C). We next extendhtoR^N×R^∗₊through the formulaw(x, ε)= Π (h∗ρε(x)). Here,Π∈C^∞(R²;R²)is such thatΠ (z)=z/|z|if|z|1/2, whileρ∈C₀^∞satisfiesρ0,

ρ=1, suppρ⊂B(0,2)\B(0,1). Then the conclusion of the theorem holds withϕj given by

ϕ₁(x):= − ∞

0

w(x, ε)∧∂w

∂ε(x, ε)dε, ϕ₂:=ϕ−ϕ₁. (4)

The inequality

|ϕ₁|W^s,pC|h|W^s,p (5)

follows from standard estimates¹for regularizationsh∗ρ_ε of mapsh∈W^s,p. As a consequence of (5), we find that

|ϕ₁|W^s,pC|g|W^s,p.

[Estimate (5) is a cousin of the estimate

x→ ∞

0

u∗ρ_ε(x)∂

∂ε

v∗ρ_ε(x) dε

W^s,p

CuL^∞|v|W^s,p, (6)

valid for any reasonable ρ and presumably well known to experts. For specialρ’s, the “discrete” and much more popular analog of (6) is the “paraproduct inequality” [7]

jk

u_jv_k W^s,p

CuL^∞|v|W^s,p, (7)

whereu=

u_j,v=

jv_j are the Littlewood–Paley decompositions ofuandv. Both (6) and (7) are refinements of the standard inequalityuvW^s,pC(uW^s,pvL^∞+ vW^s,puL^∞).]

The starting point for estimatingDϕ2is the identity

Dϕ2(x)= −2 ∞

0

∂

∂εw(x, ε)∧Dxw(x, ε)dε. (8)

At least formally, this identity follows from

Dϕ₂(x)=Dϕ(x)+ ∞

0

D_xw(x, ε)∧ ∂

∂εw(x, ε)dε+ ∞

0

w(x, ε)∧ ∂

∂εD_xw(x, ε)dε

1 One of the key estimates used in the proof is R^N_∞

0 ε^p−sp−1|D[h∗ρε](x)|^pdεdxC|h|^p_Ws,p, well known to experts [11], II.12.

=w(x, ε)∧D_xw(x, ε)_|ε=0+ ∞

0

D_xw(x, ε)∧ ∂

∂εw(x, ε)dε+w(x, ε)∧D_xw(x, ε)|^ε_ε^=∞₌₀

− ∞

0

∂

∂εw(x, ε)∧D_xw(x, ε)dε= −2 ∞

0

∂

∂εw(x, ε)∧D_xw(x, ε)dε.

Adapting an idea from [4], this identity implies the estimateDϕ2L^spC|e^ıϕ|^1/s_W^s,p. 2 The interested reader will find the detailed proof in [9].

We end this section by comparing our decomposition to the Bourgain–Brezis one. Assuming, for the sake of the simplicity, thatϕandgare defined on the wholeR^N, the decomposition in [2] is (withg=

gjthe Littlewood–Paley decomposition ofg)

ϕ1:=

jk

g_j∧g_k, ϕ2:=ϕ−ϕ1. (9)

In the same way (6) is related to (7), one may interpret the formula definingϕ₁in (9) as a discrete analog of

x→ − ∞

0

g∗ρ_ε(x)∧ ∂

∂ε

g∗ρ_ε(x) dε.

Thus our decomposition is a continuous version of the one in [2], with the additional sophistication that the regular- izations ofgare “almost projected” ontoS¹(viaΠ).

4. Some applications

As a first application, we may achieve the description ofX^s,p=C^∞(C;S¹)^Ws,p, partly obtained in [3] and [5].

Theorem 2.Let0< s <∞,1p <∞. Then

(a) ([3]) Whensp <1orspN,X^s,p=W^s,p(C;S¹);

(b) ([5]) Whens1andsp2,X^s,p=W^s,p(C;S¹);

(c) ([5]) Whens1and1sp <2,X^s,p= {e^ıϕ;ϕ∈W^s,p∩W^1,sp(C,R)};

(d) When0< s <1and1< sp < N,X^s,p= {e^ıϕ;ϕ∈(W^s,p+W^1,sp)(C,R)};

(e) When0< s <1,N2andsp=1,X^s,p= {e^ıϕ;ϕ∈(W^s,p+W^1,1)(C,R)} ∩W^s,p(C;S¹).

The lack of symmetry between statements (d) and (e) is explained by the fact that, whensp >1 ands <1, we have ϕ∈W^1,sp⇒e^ıϕ∈W^s,p; this implication fails whensp=1 ands <1. For the proof of (d) and (e), we send the reader to [9].

In a forthcoming joint paper with H. Brezis and H.-M. Nguyen [6], we investigate several consequences of Theo- rem 1. We end this Note by mentioning one of them.

Theorem 3.(See [6].) Assume thatN3,0< s <1,2sp < N,k∈ {2,3, . . .}. Then, for eachu∈W^s,p(C;S¹), there is somev∈W^s,p(C;S¹)such thatu=v^k.

This answers a question of F. Bethuel and D. Chiron [1]. [For the other values ofs,pandN, the surjectivity of the mapv→v^kis clarified in [1]. The case considered in Theorem 3 is the only one left open in [1].]

Acknowledgements

The author thanks H.-M. Nguyen for sending him an early version of [10]. He warmly thanks P. Bousquet, H. Brezis and H.-M. Nguyen for useful discussions.

References

[1] F. Bethuel, D. Chiron, Some questions related to the lifting problem in Sobolev spaces, in: H. Berestycki, M. Bertsch, F. Browder, L. Nirenberg (Eds.), Perspectives in Nonlinear Partial Differential Equations, in: Contemporary Mathematics, vol. 446, Amer. Math. Soc., 2007, pp. 125–

152.

[2] J. Bourgain, H. Brezis, On the equation divY=f and application to control of phases, J. Amer. Math. Soc. 16 (2003) 393–426.

[3] J. Bourgain, H. Brezis, P. Mironescu, Lifting in Sobolev spaces, J. Anal. Math. 80 (2000) 37–86.

[4] J. Bourgain, H. Brezis, P. Mironescu, Lifting, degree, and distributional Jacobian revisited, Commun. Pure Appl. Math. 58 (2005) 529–551.

[5] H. Brezis, P. Mironescu, On some questions of topology forS¹-valued fractional Sobolev spaces, Rev. R. Acad. Cien., Serie A Mat. 95 (2001) 121–143.

[6] H. Brezis, P. Mironescu, H.-M. Nguyen, in preparation.

[7] J.-Y. Chemin, Fluides parfaits incompressibles, Astérisque 230 (1995).

[8] P. Mironescu, Sobolev maps on manifolds: degree, approximation, lifting, in: H. Berestycki, M. Bertsch, F. Browder, L. Nirenberg, L.A.

Peletier, L. Véron (Eds.), Perspectives in Nonlinear Partial Differential Equations, In honor of Haïm Brezis, in: Contemporary Mathematics, vol. 446, Amer. Math. Society, 2007, pp. 413–436.

[9] P. Mironescu, Lifting ofS¹-valued maps in sums of Sobolev spaces, J. European Math. Soc., submitted for publication.

[10] H.-M. Nguyen, Inequalities related to liftings and applications, C. R. Acad. Sci. Paris, Ser. I 346 (17–18) (2008) 957–962.

[11] H. Triebel, Theory of Function Spaces, Birkhäuser Verlag, 1983.